Caratheodory Extension Theorem

Definition

If $\mathcal{A}$ is a Semiring on $\Omega$, and the function (pre-measure) $\tilde{m}:\mathcal{A}\to [0,\infty ]$ satisfies the following conditions, then the function $\tilde{m}$ uniquely extended to the Measure on $(\Omega ,\sigma (\mathcal{A}))$.

Conditions

• $\tilde{m}(\varnothing )=0$
• additivity: $\tilde{m}(\underset{i=1}{\overset{n}{⨄}}{B}_i)={\sum }_{i=1}^n\tilde{m}(B_i)$
• $\sigma$-sub additivity: $\tilde{m}(\bigcup _{i=1}^{\infty }{A}_i)\le {\sum }_{i=1}^{\infty }\tilde{m}(A_i)$
• $\sigma$-finite: $\exists \{A_i{\}}_{i=1}^{\infty }\subset \mathcal{A},\text{s.t.}m(A_i)<\infty ,\bigcup _{i=1}^{\infty }{A}_i=\Omega$

The conditions 1~3: $\tilde{m}$ satisfies measure-like conditions on $\mathcal{A}$ The condition 4: $\tilde{m}$ satisfies Sigma-Finite Measure-like condition on $\mathcal{A}$, it makes the extension result unique.

The set $\mathcal{A}$ should be Semiring. So, the conditions of this theorem are stronger than Extension from Pi-System, but can be applied for every Measure.